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Studia Logica (2011) 97: 183198

DOI: 10.1007/s11225-010-9303-1 Springer 2010

Graham Priest First-Order da Costa Logic

Abstract. Priest (2009) formulates a propositional logic which, by employing the world-

semantics for intuitionist logic, has the same positive part but dualises the negation, to

produce a paraconsistent logic which it calls Da Costa Logic. This paper extends matters

to the first-order case. The paper establishes various connections between first order da

Costa logic, da Costas own C, and classical logic. Tableau and natural deductions

systems are provided and proved sound and complete.

Keywords: Da Costa Logic, quantified intuitionist logic, paraconsistent logic.

1. Introduction

Dualising Intuitionist Negation1 was a paper written to celebrate the out-standing contribution of Newton da Costa to logic and many other spheresof intellectual life. It is a pleasure to add to the celebration in this paper.

Da Costas C systems of paraconsistent logic, and especially C, canbe thought of as taking intuitionist logic, preserving its positive part, but

dualising its negation to give a logic with truth value gluts, rather thangaps. There are several ways in which one might do something whichrealises this motivation.2 DIN gives a very natural way, different from daCostas, based on the Kripke semantics for intuitionist logic. Essentially, theidea is to replace the truth conditions of negation in a Kripke model:3

w() = 1 iff for all w such that wRw, w() = 0

by their dual:

w() = 1 iff for some w such that wRw, w() = 0

DIN calls the result da Costa logic.DIN deals only with propositional da Costa logic, but in Section 3 of

that paper I noted that the technique employed there could be extended

1Priest (2009). Hereafter, DIN.2

See, e.g., Brunner and Carnielli (2005), Querioz (1998), Urbas (1996).3The notation is that of DIN.

Special Issue: The Legacy of Newton da CostaEdited by Daniele Mundici and Itala M. Loffredo DOttaviano

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184 G. Priest

to quantifiers. This note spells out the details. A first-order version of thepropositional semantics is, in fact, given by Rauszer (1977). However, un-like Kripke interpretations for intuitionist logic, this uses constant domains,

and hence verifies principles that are not valid in positive intuitionist logic,notably the confinement principle: x(P x Qa) xP x Qa.4

First, I will specify a Kripke semantics for first-order da Costa logic.Next, I will give a tableau system which is sound and complete with respectto the semantics. This is then used to establish various facts about the logic,its relationship to classical logic, C, and intuitionist logic. Finally, I give anatural deduction system which is sound and complete with respect to thesemantics.5

2. Kripke Semantics

Let us start with the Kripke semantics for intuitionist logic. These are asin 20.3 of Introduction to Non-Classical Logic6though I have made twonecessary changes to the presentation there. First, I have included a desig-nated world, @. The semantics (and definition of validity) of INCL couldhave been formulated equivalently in this way. In INCL every constant ofthe language is required to denote something in the domain of every world.This is no longer appropriate once we move to da Costa logic, since con-stants that denote something at the base world may denote things thatdo not exist once we move backwards (in time) down the relation R. Thismakes the alternative approach mandatory. Secondly, I have required thatw(P) Dw, not D. The constraint is standard in formulations of intuition-istic logic. (If one has established that an object, a, has a property, thena must have already been constructed.) However, it makes no difference tothe valid inferences.7 It does in the present case, as we shall have occasion

to note.A Kripke semantics for first-order intuitionist logic is as follows. An in-

terpretation is a structure W, @, R , D , , where W is a non-empty set ofworlds, and @ is a distinguished member of W. R is a reflexive and tran-

4Rauszers logic also contains the dual of the intuitionist . This can be added to thelogic of the present paper as well, as in DIN, Section 7.

5DIN also investigates the algebraic semantics for propositional da Costa logic. Theseextend naturally to a semantics for first-order da Costa logic in terms of complete da Costa

algebras (the duals of complete Heyting algebras). I leave the details of this for anotheroccasion.

6Priest (2006), hereafter INCL.7See INCL, 20.13, Problem 12.

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First-Order da Costa Logic 185

sitive relation on W. For every w W, Dw is a collection of objects, andD@ = . D =

{Dw : w W}. is a function such that for every constantin the language, a, (a)

D@;

8 and for every world, w, and n-place predi-

cate, P, w(P) Dw. (This is the so called Negativity Constraint, since it isthe hallmark of negative free logics.9) The structure satisfies the followingconditions, for all w, w, and P:

if wRw then Dw Dwif wRw then w(P) w(P)

A truth value, w()

{1, 0

}, is assigned to every formula, , at every

world, w, by the following truth conditions:10

w(P a1 . . . an) = 1 iff(a1), . . . , (an) w(P)w( ) = 1 iff w() = 1 and w() = 1w( ) = 1 iff w() = 1 or w() = 1w( ) = 1 iff for all w such that wRw, if w() = 1 then w() = 1w(

) = 1 iff for all w such that wRw, w() = 0

w(x) = 1 iff for some d Dw, w(x(kd)) = 1w(x) = 1 iff for all w such that wRw, and all d Dw , w(x(kd)) = 1

The truth conditions and the constraints on the structure ensure theHeredity Condition:

if wRw and w() = 1 then w() = 1

for all w and . (The proof is by a simple induction over the constructionof formulas.)

Validity is defined in terms of truth preservation at @ in all interpreta-tions:

iff, for all interpretations, if @() = 1 for all , @() = 1

8The truth conditions of quantifiers below require further constants. The denotations

of these constants have to be in D, but not D@.9INCL, 13.4.2.10kd is a constant such that (kd) = d, and x(a) is with all occurrence ofx replaced

by a.

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186 G. Priest

The semantics for da Costa logic are exactly the same as those for intu-itionist logic, except that the truth conditions for negation are replaced bytheir dual:

w() = 1 iff for some w such that wRw, w() = 0It is easy to establish that the Heredity Condition is still forthcoming, andhence that the logic is closed under uniform substitution (since any formulathen behaves as does a propositional parameter).11

Da Costa logic and intuitionist logic have the same positive part; and thenegation of each conservatively extends this (since any positive interpretationcan be extended trivially to an interpretation for either kind of negationor

both). The propositional part of da Costa logic is as in DIN. The noveltyof first-order da Costa logic is therefore in the interplay between negationand the quantifiers. We will consider this in a moment. First, it will help tohave a tableau system for the logic and its associated machinery.

3. Tableaux

A tableau system for intuitionist logic is as follows.12 There is a designatedexistence predicate, E, thought of as applying at world w, to the things inDw. Lines are of the form , +i, , i or irj. A tableau for the inference1, . . . , n/ starts with lines of the form 1, +0, . . . , n, +0, , 0, as well asEa, +0, for every constant, a, occurring in the premises and conclusionorone of the form Ec, +0 if there are none such.

A branch of a tableau closes if it contains lines of the form , +i and , i.The tableau itself closes if all branches do. The rules for the connectives areas follows:

, +i

, +i, +i

, i

, i , i

11The proof is by induction over the construction of formulas. The cases for the con-nectives are as in DIN. The case for the particular quantifier is left as an exercise. Hereis the case for the universal quantifier. Suppose that wRw and w (x) = 1. Then forsome w such that wRw, and some d Dw , w (x(kd) = 0. Since R is transitive,w(x) = 0.

12

This corresponds to the tableaux of type 1, INCL, 20.4. 20.5 gives an alternativetableau system, type 2, which does not employ an existence predicate. It is harder toformulate type 2 tableaux in the present case, since, because of the Negativity Constraintand the backward looking nature of negation, Dw is determined by more than just thequantifier rules.

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First-Order da Costa Logic 187

, +i

, +i , +i

, i

, i, i

, +iirj

,

j , +i

, i

irk, +k, k

, +iirj

, j

, i

irk, +k

where k, in any of these rules, is a number new to the branch. We also haverules for r and the Heredity Condition:

. irj irj jrk P a1 . . . an, +i

iri irk P a1 . . . an, +j

where P is any predicate (including E). For quantifiers, the rules are as

follows: xA, +i

Ec, +iAx(c), +i

xA, i

irkEc, +k

Ax(c), kxA, i

Ea, i Ax(a), i

xA, +iirj

Ea, j Ax(a), +j

c and k are new to their branches. a is any constant on the branch.

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188 G. Priest

For da Costa logic, we simply replace the rules for negation with:

, +i

kri

, k

,

i

jri, +j

where k is new to the branch; and add the Negativity Rule:

P a1 . . . an, +i

Ea1, +i

...Ean, +i

where P is any predicate (except E, for which the rule is redundant).Example 1. Here is a tableau to show that xP x xP x

Ec, +0xPx, +0xP x, 0

0r01r0, 1r1

xPx, 10r2, 2r2, 1r2Ea, +2

P a, 2P a, +2P a, +0P a, +1Ea, +1Ea, +0

Ea, 1 P a, 1

Lines 5 and 6 follow from line 2. Lines 7, 8, and 9 follow from line 3. The

next three lines follow from line 9 and the facts about r. The next two linesfollow from the Negativity Rule. The lines after the split follow from line 6.Note that without the Negativity Rule, the left branch of the tableau wouldnot close. An interpretation can be read off from the open branch, thus

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First-Order da Costa Logic 189

modified, demonstrating that the inference is invalid without the NegativityConstraint, which shows that the constraint affects validity in a way that itdoes not in intuitionist logic. Details are left as an exercise.

Example 2. Here is another tableau, to show that: xP x xP x

Ec, +0xP x, +0xPx, 0

0r0Ea, +0

P a, +0

1r0, 1r1P a, 1

xPx, +0xPx, +1

Ea, 0 P a, +0

Ec, 0 P c, +0

Ea, 1 P a, +1

Ec, 1 P a, +1

Ea, +1

Ec, 1 P c, +1

Ea, +1Ec, +1

Lines 5 and 6 follow from line 2. Lines 7 and 8 follow from line 6. Lines 9

and 10 follow from line 3. The various splits then follow from lines 9 and 10.The final formulas concerning E on the right hand branches follow by theNegativity Rule.

Given an open tableau, a countermodel is read off from an open branch asfollows. There is a world, wi, for every i on the branch. @ is w0. wiRwj iffirjis on the branch. (b) = b, and (a1), . . . , (an) wi(P) iff P a1 . . . an, ioccurs on the branch. Dwi = wi(E).

In the countermodel determined by the left hand open branch of the

above tableau: W = {w0, w1}, @ = w0, w0Rw0, w1Rw1, and w1Rw0.w0(E) = Dw0 = {a, c} and w1(E) = Dw1 = . (a) = a, (c) = c,w0(P) = {a, c} and w1(P) = . We may depict the interpretationas follows.

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190 G. Priest

w1

w0 (= @)

a cP E

a cP

E

P a fails at w1, so P a holds at w0. Since a exists at w0, xP x is true atw0. xP x is true at w1 and w0; hence xP x fails at w0.

The tableau system is sound and complete with respect to the semantics.The proof is a straightforward modification of that given for intuitionist

tableaux of type 1 in INCL. We establish two lemmas.13

Locality Lemma Let I1 = D,W,R,1, I2 = D,W,R,2 be two interpreta-tions. Let the language of the interpretations be L. If is any closed formulaof L such that 1 and 2 agree on the denotations of all the predicates andconstants in it, then, for all w W:

1w() = 2w()

Denotation Lemma Let I = D,W,R, be any interpretation. Let itslanguage be L. Let be any formula of L with at most one free variable,x, and a and b be any two constants such that (a) = (b). Then, for allw W:

w(x(a)) = w(x(b))

The proofs of the lemmas are as in INCL 20.9.2 and 20.9.3, except thatthe cases for

are different in the obvious ways. The Soundness and Com-

pleteness proofs are then as in 23.9.4-23.9.9, except that (i) the clauses fornegation in the Soundness and Completeness Lemmas are as in DIN, Sec-tion 3; and (ii) in the induced interpretation, @ = w0, and the interpretationsatisfies the Negativity Constraint because of the Negativity Rule.

4. The Properties of da Costa Logic

Let us now turn to the properties of first-order da Costa logic. We start

with the relationship between da Costa logic and C.

13The language of an interpretation is the language augmented by a constant, kd, foreach d D.

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First-Order da Costa Logic 191

DIN, Section 2, shows that propositional da Costa logic properly extendspropositional C. Axiomatically, first-order C is obtained from proposi-tional C by adding the following axioms and rules:

14

x(c) x x x(c)If x(c) then xIf x(c) then x

where c does not occur in . We may establish that these are valid/validity-preserving in da Costa logic as follows.

For the first axiom, suppose that x(c) x. Then there is aninterpretation, I, such that x(c) x fails at @. Hence, for some w suchthat @Rw, x(c) holds at w, and x does not. Then for every d Dw,x(kd) fails at w. But (c) D@, and so (c) Dw. Where d = (c), theDenotation Lemma therefore entails that x(c) fails at w. Contradiction.

15

For the first rule, suppose that x. Then there is an interpreta-tion where x fails at @. So there is a w such that @Rw, is true atw, and

x fails at w. Hence, for some w such that wRw and some d

Dw ,

x(kd) fails at w. By heredity, holds at w. Now take an interpretationwhich is the same as I, except that (c) = d. In this interpretation, by theDenotation Lemma and the Locality Lemma, holds at w and x(c) failsat w. Since @Rw, x(c) fails at @ in this interpretation. Hence it isnot logically valid.

The cases for the second axiom and the second rule are similar. Conse-quently, first-order da Costa logic (properly) extends first-order C.

Next, classical logic. Any inference valid in first-order da Costa logic is

valid in classical logic (since a classical countermodel is, effectively, a one-world da Costa interpretation). The converse is not true, since da Costa logicis paraconsistent. Despite this, the logical truths of the -free fragmentof propositional da Costa logic coincide with the -free logical truths ofclassical logic, as proved in DIN, Section 2.

However, for the first-order case, not even the logical truths of the --- or the --- fragments are identical with those of classical logic. The

14Da Costa (1974: 503).15Note that the fact that x(a) is true at a world does not, in general, entail that x

is true there. Take an interpretation with two worlds, w and @, where wR@, wRw, @R@,D@ contains just the the denotation of a, and Dw is empty. Then Pa is true at w, butxPx is not true there.

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192 G. Priest

first of these is perhaps not so surprising given the domain-shift in the truthconditions of. Indeed, a variation of the confinement principle fails: here isa countermodel to show that the classically valid

x(P a

Qx)

(P a

xQx)

is not valid in da Costa logic:

w0 (= @) w1a b

P Q

E

a bP

Q

E

P a fails at w0, and xQx fails at w0 (since Qb fails at w1). Hence, P axQxfails at w0. But P a Qa holds at w0 and w1, and P a Qb holds at w1.Hence x(P a Qx) holds at w0, so x(P a Qx) fails at w0 too.

For the second, here is a countermodel to show that x(P x P x),which holds in classical logic:

w0(= @)

a0 a1 a2 . . .

P . . .E

. . .

w1

a0 a1 a2 . . .P . . .E

. . .

w2

a0 a1 a2 . . .P . . .E

. . .

...

P a0 P a0 holds at w0; P a1 P a1 holds at w1; and so on. Hence, x(P xP x) holds at every world, and so x(P x P x) fails at w0.

Finally, the relationship between the quantifiers and negation in da Costa

logic. Consider the following:

1. x x2. x x

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First-Order da Costa Logic 193

3. x x4. x x

In intuitionist logic, 1, 3, and 4 are true; 2 is not. One might have expectedthat in da Costa logic, the relationship would be dual: that we would have3 and 4, and 2 but not 1or maybe 1 and 2, but only one of 3 and 4. Butin fact, all of 1-4 fail (though a special case of 4 holds, as we have seen inExample 1 of the previous section). We have already seen that 1 is invalid inExample 2 of the previous section. Here are countermodels for 2, 3, and 4.

For 2, xP x xP x:

w0 (= @)

w2a c

P E

a cP E

w1a c

P E

Because of w2, xP x fails at w1; hence xP x holds at w0. P c holds at w0and w1; hence P c fails at w0, as, therefore, does xP x.

For 3, xP x xP x:

a cP

E

w1

w0(= @)

a cP

E

a c

P

E

w2

P a and P c hold at w0, as, then, does xP x. xP x is true at all worlds;hence, xP x fails at w0.

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194 G. Priest

Finally, and a bit less obviously, for 4, xP x xP x:

w1 w0(= @)a

P E

aP E

xP x fails at w1 (since nothing exists there). Hence xP x holds at w0.But P a fails at w1. Hence P a holds at w1 and w0; and so P a fails atw0. Consequently, xP x fails at w0.

5. Natural Deduction

Tableau proof systems are very good for determining validity and invalidity,but for many purposes a natural deduction system is betterespecially sincethe tableau system given above makes use of an existence predicate, whichis alien to most presentations of intuitionist logic. So let us, finally, see howa natural-deduction system for first-order da Costa logic may be obtainedfrom one for positive intuitionist logic. The system is in the style of Prawitz.

The rules for positive intuitionist logic are as follows:16

...

...

...

x x(a)x(a) x

16See, e.g., Tennant (1978), ch. 4.

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First-Order da Costa Logic 195

In the second rule for , a does not occur in or in any undischargedassumption on which x(a) depends.

x(a)

x

x(a)...

x

In the second rule for , a does not occur in , , or in any undischargedassumption on which depends (other than x(a)).

For da Costa logic, we add to these the following rules for negation:17

.

In the second of these, the deduction of has no undischarged assump-tions.

The deduction system is sound and complete with respect to the seman-tics. For soundness, we establish that if we have a deduction of withundischarged assumptions then . This is established by a recur-

sion over the structure of proofs. The cases for the propositional operatorsare essentially as in DIN, Section 5. The cases for the first of each pair ofquantifier rules are straightforward. Here are the cases for the second ofeach pair.

Suppose that we have a proof ofx(a) with undischarged assumptions .By induction hypothesis, x(a). Consider any interpretation, I, in whichall members of are true at @. Let d D@, and let I be the interpretationof the language which is the same as I, except that (a) = d. By the LocalityLemma, all the members of are true at @ in this interpretation. Hence

x(a) is true at @ in I. By the Denotation Lemma, x(kd) is true at @.Hence, x is true at @ in I. By the Locality Lemma, x is true at @ inI, as required.

Suppose that we have a proof of x with undischarged assumptions1, and a proof of with undischarged assumptions 2 {x(a)} (where2 does not contain x(a)). By induction hypothesis, 1 x and 2 {x(a)} . Now suppose that all members of 1 2 are all true at @in some interpretation, I. Then x is true at @, and so for some d D@,x(kd) is true at @. Let I

be the same as I, except that (a) = kd. Thenby the Locality Lemma, all members of 2 are true at @ in I

, and by

17DIN, Section 5.

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196 G. Priest

the Denotation Lemma x(a) is too. Hence is true at @ in I. By the

Locality Lemma, is true at @ in I, as required.The Completeness proof is an extension of that for the propositional

case in DIN, Section 5. Call a set of sentences, , saturated in a set ofconstants, C, iff whenever x , x(c) , for some c C. Call prime deductively closed and saturated (pdcs) in C iff it is prime (that is, if then or ), deductively closed, and saturated in C.

Fix some uncountable set of constants, K. We now have the follow-ing fact:18

Lemma 1 (Fundamental Lemma). If there is set , pdcs in a

countable subset of K, such that and .Proof. Let C be a countable subset of K disjoint from the constants in . Enumerate the formulas of the language of augmented by C:0, 1, . . . Define the following sequence of sets by recursion:

0 =

n+1 = n {n} if n {n} , and n otherwise. = i <

i

As in the propositional case, is prime, deductively closed, and . is also saturated in C. For suppose otherwise. Then for some x ,there is no c C such that such that x(c) . Let x be i. Only afinite number of members of C occur in i+1. Let c be one that does not.Then x(c) / . x(c) is j for some j > i. Hence j {x(c)} . Soj {x} . Since x j, j , which is not the case.

Now suppose that . We define a countermodel as follows. W =

{ : for some countable C K, is pdcs in C}. R iff . If W,let C be the set of constants occurring in . (Clearly, is saturated inC.) D = C. (a) = a, (P) = {a1,...,an : P a1 . . . an }. Finally,by the Fundamental Lemma, there is some such that and (so that / ). @ = .

It is easy to check that this structure is an interpretation for da Costalogic. The only point that is not immediate concerns the domain-increasingcondition. Suppose that c C and R. Let x be any logical truthwithout constants. Then x(c) . Since , x(c) . Hence,D D.

18 means that for some n 1, and 1, . . . , n , 1 . . . n.

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First-Order da Costa Logic 197

We now need one more lemma:

Lemma 2 (-Lemma). If W and x / , then there is a W suchthat R, and for some c D, x(c) / .Proof. Let = . Let c be some constant not in {}. Without lossof generality, we can take it to be the first constant in the set C used inthe proof of the Fundamental Lemma. Let = {x(c)}. Now, .For suppose that x(c). Then since c has no occurrences in {}, x, and so x , which it is not. So by the Fundamental Lemma,there is a W, with c D, such that and x(c). This hasthe required properties.

To finish the proof, we need to show that for any W, and any inthe language of :

() = 1 iff

The theorem then follows, since the interpretation defined is a countermodel.The proof is by recursion. The atomic case is true by definition. The

cases for the propositional connectives are as in DIN, Section 5. This leave

the quantifiers.

If (x) = 1 then for some c C, (x(kc)) = 1. But (kc) = c.So by the Denotation Lemma, (x(c)) = 1. By IH, x(c) . Hencex , by deductive closure.

Conversely, suppose that x . Then for some c C, x(c) .By IH, (x(c)) = 1. By the Denotation Lemma, (x(kc)) = 1; so(x) = 1.

Suppose that x . Then for all such that R, x . Bydeductive closure, for all c C, x(c) . By IH and the DenotationLemma, for every c D, (x(kc)) = 1. Hence, (x) = 1.

Conversely, suppose that x / . The by the -Lemma, there is a W such that R and for some c D x(c) / . By IH and theDenotation Lemma, (x(kc)) = 1. So (x) = 1.

6. Conclusion

Da Costa logic is a smooth logic which extends positive intuitionist logic in anatural way, both semantically and proof-theoretically. The first-order ver-sion extends the propositional version exactly as one would expect. Perhaps

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198 G. Priest

the most unexpected fact about it is provided by the interaction betweennegation and the quantifiers. One might have expected the duals of the re-lations that obtain in intuitionist logic to obtain. However, the backward

looking nature of negation interacts with the forward-looking quantifiertruth-conditions and the domain-increasing feature of the semantics to en-sure that this is not the case.

References

[1] Brunner, A., and W. Carnielli, Anti-intuitionism and Paraconsistency, Journal of

Applied Logic 3: 161184, 2005.

[2] da Costa, N., On the Theory of Inconsistent Formal Systems, Notre Dame Journal

of Formal Logic 4: 497510, 1974.[3] Priest, G., Introduction to Non-Classical Logic: From If to Is, Cambridge: Cambridge

University Press, 2006.

[4] Priest, G., Dualising Intuitionist Negation, Principia 13: 16589, 2009.

[5] Rauszer, C., Applications of Kripke Modes to Heyting-Brouwer Logic, Studia Logica

36: 6171, 1977.

[6] Tennant, N., Natural Logic, Edinburgh: Edinburgh University Press, 1978.

[7] Querioz, G., Sobre a Dualidade entre Intuitionismo e Paraconsistencia, PhD thesis,

State University of Campinas, 1998.

[8] Urbas, I., Dual Intuitionist Logic, Notre Dame Journal of Formal Logic 37: 440451,

1996.

Graham Priest

Departments of PhilosophyUniversities of Melbourne and St AndrewsAustraliaand

the Graduate Center

City University of New York, USAg.priest@unimelb.edu.au